Let $X=\{11,12,13, \ldots ., 40,41\}$ and $Y=\{61,62$, $63, \ldots ., 90,91\}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and $\sigma^2$ is the variance of all the observations in $X \cup Y$, then $\left|\overline{ x }+\overline{ y }-\sigma^2\right|$ is equal to $……………..$.

The mean of two samples of size $200$ and $300$ were found to be $25, 10$ respectively their $S.D.$ is $3$ and $4$ respectively then variance of combined sample  of size $500$ is :-

Let $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{N}$ and $\mathrm{a}<\mathrm{b}<\mathrm{c}$. Let the mean, the mean deviation about the mean and the variance of the $5$ observations $9$,$25$, $a$, $b$, $c$ be $18$,$4$ and $\frac{136}{5}$, respectively. Then $2 \mathrm{a}+\mathrm{b}-\mathrm{c}$ is equal to …………..

A data consists of $n$ observations

${x_1},{x_2},……,{x_n}.$ If $\sum\limits_{i – 1}^n {{{({x_i} + 1)}^2}}  = 9n$ and $\sum\limits_{i – 1}^n {{{({x_i} – 1)}^2}}  = 5n,$ then the standard deviation of this data is

The mean and standard deviation of $100$ observations were calculated as $40$ and $5.1$ , respectively by a student who took by mistake $50$ instead of $40$ for one observation. What are the correct mean and standard deviation?